Differential cross section measurements in particle physics experiments are smeared by the finite resolution of the particle detectors. Using the smeared observations to infer the true particle-level spectrum is an ill-posed inverse problem, typically referred to as unfolding or unsmearing. The defining feature of this problem is that it is easy to go from the particle-level spectrum to the smeared observations but the inverse direction of inferring the quantity of interest based on the smeared data tends to produce extremely unstable solutions. It is customary to address this using regularization which reduces the variance of the estimates at the expense of increased bias. While this can lead to well-behaved point estimates, it is extremely challenging to provide rigorous frequentist uncertainties for the regularized estimates. In this talk, I will first give an overview of the statistical techniques that are commonly used for regularized unfolding at the LHC. I will then demonstrate that some of these methods may seriously underestimate the statistical uncertainty and will explain why that is the case. I will then describe approaches that can be used to obtain improved frequentist uncertainty quantification in unfolding. I will argue that the key is to avoid explicit regularization and instead infer functionals of the unknown spectrum that implicitly regularize the problem.